Weak laws of large numbers for sequences of random variables with infinite \(r\)th moments (Q1714988)

The authors use a wider definition of stochastic domination. A sequence of random variables \((X_n,n\geq 1)\) is said to be stochastically dominated by the random variable \(X\) if, for some finite \(C>0\), \(\sup_nP(| X_n| \geq t)\leq CP(| X| \geq t)\), for all \(t\geq 0\). Theorem 5: Let \(0<r<2\) and \((X_n,n\geq 1)\) be a sequence of independent random variables. Suppose that \((X_n,n\geq 1)\) is stochastically dominated by the random variable \(X\) and \(H(x)=E(| X| ^rI(| X| <x))\) is a slowly varying function at infinity. Let \((c_{nk}; 1\leq k\leq m_n,n\geq 1)\) be a triangular array of real numbers, such that \(\sup_n\sum_{k=1}^{m_n}| c_{nk}| ^rH(| c_{nk}| ^{-1})<\infty\) and \(\max_{1\leq k\leq m_n}| c_{nk}| \to 0\) as \(n\to\infty\). If \(0<r\leq 1\), then \(\sum_{k=1}^{m_n}c_{nk}(X_k-E(X_kI(| c_{nk}X_k| \leq 1)))\to 0\) in probability as \(n\to\infty\). If \(1<r<2\) then \(\sum_{k=1}^{m_n}c_{nk}(X_k-E(X_k))\to 0\) in probability as \(n\to\infty\). Theorem 11 establishes weak laws of large numbers for a weighted sum of independent random variables with \(r\)th order decay of tail probability if \(0<r<2\). It is shown that these results imply a weak law of large numbers of extended Pareto-Zipf distributions and generalized Feller game. The results are also applied to study weak law of large numbers of moving average sums of a sequence of i.i.d. random variables.